Geoscience Reference
In-Depth Information
∂
U
∂
+
U
∂
U
∂
+ g
∂ξ
∂
= 0
,
(5.47)
t
x
x
(
x
)
U
= 0
,
∂ξ
∂
+
∂
∂
ξ
−
α
(5.48)
t
x
0
and
period
T
. We introduce dimensionless variables (the asterisk '*' will be further
dropped)
Consider the wave, arriving on the shelf, to be characterized by a height
ξ
t
T
,
x
∗
=
x
α
ξ
0
,
ξ
∗
=
ξ
U
∗
=
U
T
ξ
0
α
t
∗
=
ξ
0
,
.
In these variables the systems (5.47) and (5.48) assume the following form
[Kaistrenko et al. (1985)]:
∂
U
∂
+
U
∂
U
∂
Br
∂ξ
1
+
= 0
,
(5.49)
t
x
∂
x
∂ξ
∂
∂
∂
+
x
[(
ξ
−
x
)
U
]=0
,
(5.50)
t
ξ
0
(g
2
T
2
) is the only dimensionless parameter, which from a physical
point of view represents a criterion for the breaking of a wave running up a plane
slope. Note that this criterion is not quite precise, since it does not take into account
phase dispersion and bottom friction.
Numerous experimental studies have permitted to introduce the Iribarren number
as a criterion for wave breaking [Battjes (1988)],
where
Br
=
α
1
/
2
Ir
=
αλ
,
1
/
2
0
ξ
where
is the deep-water wavelength. We consider the depth along the slope
to increase indefinitely, therefore, for waves of any length there exists a region,
where they do not 'feel' the bottom. Expressing the wavelength via the period from
the dispersion relation for gravitational waves in deep water,
λ
= g
T
2
(2
),we
obtain, that the empirically introduced Iribarren parameter and the quantity
Br
are
uniquely related to each other,
Ir
−
2
= 2
λ
π
π
Br
. The existence of such a relationship
testifies in favour of the correct choice of non-linear long-wave model for describing
the tsunami run-up on the shore. Transition from surging to plunging breaker (wave
breaking) occurs when
Ir
0
.
04).
An important step in resolving the run-up problem was the work [Carrier,
Greenspan (1958)], in which it was shown that non-linear long-wave equations can
be reduced to a linear wave equation, which, unlike the initial system is resolved in
semispace with a fixed boundary. We recall that the initial system has an unknown
movable boundary—the shoreline. This transformation was subsequently termed
the Carrier-Greenspan transformation.
The approach based on the Carrier-Greenspan transformation has permitted to
find a whole series of analytical solutions to the problem of tsunami run-up on
a plane slope (e.g. [Pelinovsky (1996)]).
≈
2(
Br
≈
